One common type of refraction is the bending of the path of a lightwave as it crosses a boundary between two media. In conventional optics, Snell's law gives the relationship between the angles of incidence and refraction for wave crossing the boundary:n1 sin(θ1)=n2 sin(θ2).
This relationship is shown in FIG. 1, where a ray 100 in a first medium 101 arrives at a boundary 102 at an angle θ1, as referenced to the normal 104. As the ray 100 crosses the boundary 102, the ray 100 is bent so that it continues propagating as a refracted ray 106 at an angle θ2.
While this relatively simple ray optics presentation of refraction is widely accepted, a more thorough examination of refraction involves consideration of propagation of electromagnetic waves and considerations of energy reflected at a boundary. FIG. 2A represents this diagrammatically with an incident wave 108 crossing a boundary 110. A portion of the energy is reflected as represented by the wave 112 and a portion of the energy propagates as a transmitted wave 114. As represented by the spacing between the waves, the wavelength of the transmitted wave 114 is shorter than that of the incident wave 108, indicating that the refractive index n2 experienced by the transmitted wave 114 is higher than the refractive index n1 experienced by the incident wave 108.
As indicated by the figures and by Snell's law, a lightwave traveling from a lower index of refraction to a higher index of refraction will be bent toward the normal at an angle determined by the relative indices of refraction. For this system, in a conventional analysis, the range of refracted angles relative to the normal is typically confined to a range from 0 degrees to a maximum angle determined by an angle of total reflection. Additionally, the amount of light energy reflected at the boundary is a function of the relative indices of refraction of the two materials.
More recently, it has been shown that under certain limited conditions, rays traveling across a boundary may be refracted on the same side of the normal as the incident ray in a phenomenon called “negative refraction.” Some background on the developments can be found in Pendry, “Negative Refraction Makes a Perfect Lens,” Physical Review Letters, Number 18, Oct. 30, 2000, 3966–3969; Shelby, Smith, and Schultz, “Experimental Verification of a Negative Index of Refraction,” Science, Volume 292, Apr. 6, 2001, 77–79; Houck, Brock, and Chuang, “Experimental Observations of a Left-Handed Material That Obeys Snell's Law,” Physical Review Letters, Number 13, Apr. 4, 2003, 137401-(1–4); each of which is incorporated herein by reference. With particular reference to negative refraction, Zhang, Fluegel and Mascarenhas have demonstrated this effect at a boundary between two pieces of YVO4 crystal, where the pieces of crystal are rotated such that the ordinary axis of the first piece is parallel to the extraordinary axis of the second piece. This demonstration was presented in Zhang, Fluegel and Mascarenhas, “Total Negative Refraction in Real Crystals for Ballistic Electrons and Light,” Physical Review Letters, Number 15, Oct. 10, 2003, 157404-(1–4), which is incorporated herein by reference. FIG. 2B shows the interface, the relative axes and the nomenclature used in the descriptions herein for the case of positive refraction. FIG. 2C shows the same aspects for negative refraction.
The YVO4 crystal treated by Zhang, et al., is an example of an anisotropic crystal whose dielectric permittivity is defined by the matrix,
         (                                        ɛ            o                                    0                          0                                      0                                      ɛ            e                                    0                                      0                          0                                      ɛ            z                                )  and where ∈o, ∈e, and ∈z are not all the same. In general, the refractive index n of a medium is related to the dielectric permittivity ∈ as,n=c√{square root over (μ∈)},where μ is the magnetic permeability of the medium.
A more general case, described by Zheng Liu, et al., NEGATIVE REFRACTION AND OMNIDIRECTIONAL TOTAL TRANSMISSION AT A PLANAR INTERFACE ASSOCIATED WITH A UNIAXIAL MEDIUM, Phys. Review B (115402), dated Mar. 4, 2004, bearing submission date Oct. 13, 2003, relates to an interface between a uniaxial medium and a second medium and is incorporated herein by reference. Liu describes the propagation of waves through uniaxial and isotropic materials to demonstrate reflectionless refraction at an interface between two uniaxial materials or between a uniaxial material and an isotropic material.